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66
ANALYTICAL MKCHANICS
The curve, Fig. 43, defined by equation (15) is called an exponential curve. It has an interesting property, namely, its ordinate is doubled every time a constant value P is added to its abscissa. This constant is called the half-value period of the curve. The value of P may be determined in the following manner. By the definition of P and from equation (15) we have
Fi
Dividing equation (16) by equation (15) we get
or                                 P = a log,, 2.
LENGTH OF CABLE.In order to find the length in terms of the span eliminate y between equations (7) and (11). This gives
/  x        _;r\
Q _ :_ I pa __p    a                                                                                        f I 7 \
o       \ t/    ^^ \j       I                                                                                     \ I i /
w       rt \                     /                                                                                      \ * * /
= X+J   5! +      1       *+ ... f       (IK)
t  o a      & * o * *i * o u
where the right member of equation (IS) is obtained by expanding the right-hand member of equation (17) by Maclaurin's Theorem.
If D and L denote the span and the length of the rubles respectively, we have s = f L when ori/>. Therefore substituting these values of s and x in equation (18) awl replacing a by its value we obtain
48
(19)